Roughly speaking, it shows that all such groups are similar to the general linear group over a field.

For example, we have the general linear groups over finite fields.

For general linear groups this was already known by the work of .

All matrix groups are subgroups of some general linear group.

Basic examples are , the general linear groups over the complex numbers.

For the general linear group, we get as the Cartan involution.

That is to say, the general linear group acts transitively on the set of all complete flags.

They form the general linear group under composition.

More generally, the matrix ring and the general linear group are locally profinite.

It does however exclude the general linear group.

Every periodic subgroup of the general linear group over the complex numbers is locally finite.

For example the general linear group of 2x2 real invertible matrices has a transitive action on it.

The variety Y naturally has an action of , a product of general linear groups.

For general linear groups over local fields, the L-packets have just one representation in them (up to isomorphism).

A basic example of a topological group with no small subgroup is the general linear group over the complex numbers.

In representation theory they are the characters of irreducible representations of the general linear groups.

The Sylow subgroups of general linear groups are another fundamental family of examples.

The group G', can be interpreted as the general linear group of a one-dimensional vector space.

Together, these constructions are used to generate the irreducible representations of the general linear group; see fundamental representation.

A "group representation" is a homomorphism from a group to a general linear group.

One such family of groups is the family of general linear groups over finite fields.

The center of the general linear group is the collection of scalar matrices .

In the case of the general linear group, all fundamental representations are exterior powers of the defining module.

If a basis for the complex vector space 'V' is chosen, the representation can be expressed as a homomorphism into general linear group.

There are infinite dimensional analogues of general linear groups, orthogonal groups, and so on.